Theorem nssinpssOLD 2825

Description: Negation of subclass expressed in terms of intersection and proper subclass.

Assertion

Ref	Expression
nssinpssOLD	|- (-. A C_ B <-> (A i^i B) C. A)

Proof of Theorem nssinpssOLD

Step	Hyp	Ref	Expression
1		inss1 2812	. . 3 |- (A i^i B) C_ A
2	1	biantrur 794	. 2 |- (-. A C_ (A i^i B) <-> ((A i^i B) C_ A /\ -. A C_ (A i^i B)))
3		ssid 2634	. . . . 5 |- A C_ A
4	3	biantrur 794	. . . 4 |- (A C_ B <-> (A C_ A /\ A C_ B))
5		ssin 2814	. . . 4 |- ((A C_ A /\ A C_ B) <-> A C_ (A i^i B))
6	4, 5	bitri 190	. . 3 |- (A C_ B <-> A C_ (A i^i B))
7	6	notbii 204	. 2 |- (-. A C_ B <-> -. A C_ (A i^i B))
8		dfpss3 2695	. 2 |- ((A i^i B) C. A <-> ((A i^i B) C_ A /\ -. A C_ (A i^i B)))
9	2, 7, 8	3bitr4i 200	1 |- (-. A C_ B <-> (A i^i B) C. A)

Syntax hints:  -. wn 2   <-> wb 163   /\ wa 240   i^i cin 2592   C_ wss 2593   C. wpss 2594

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-in 2603  df-ss 2605  df-pss 2607

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